Heat Transfer in Fully-Developed Internal Turbulent Flow

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Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (<math>{{{q}''}_{w}}=</math> const) will be considered in this subsection (Oosthuizen and Naylor, 1999). When the turbulent flow in the tube is fully developed, we have <math>\bar{v}=0</math> and the energy eq. (5.268) becomes

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Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (<math>{{{q}''}_{w}}=</math> const) will be considered in this article <ref name="ON1999">Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York.</ref><ref>Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.</ref>. When the turbulent flow in the tube is fully developed, we have <math>\bar{v}=0</math> and the energy equation becomes

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|{{EquationRef|(2)}}

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where <math>{{\bar{T}}_{c}}</math> is the time-averaged temperature at the centerline of the tube, and Tw is the wall temperature. Thus,

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where <math>{{\bar{T}}_{c}}</math> is the time-averaged temperature at the centerline of the tube, and ''T<sub>w</sub>'' is the wall temperature. Thus, <math>({{T}_{w}}-\bar{T})/({{T}_{w}}-{{\bar{T}}_{c}})</math> is a function of ''r'' only, i.e.,

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<math>({{T}_{w}}-\bar{T})/({{T}_{w}}-{{\bar{T}}_{c}})</math>

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is a function of r only, i.e.,

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where f is independent from x. Differentiating (5.295) yields

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where ''f'' is independent from ''x''. Differentiating eq. (2) yields

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|{{EquationRef|(5)}}

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Substituting eq. (5.296) into eq. (5.298), one obtains:

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Substituting eq. (3) into eq. (5), one obtains:

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Since the heat flux is constant, <math>{{{q}''}_{w}}=</math> const, it follows that

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Since the heat flux is constant, <math>{{{q}''}_{w}}=</math> const, it follows that <math>({{T}_{w}}-{{\bar{T}}_{c}})= Const</math>, i.e.,

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<math>({{T}_{w}}-{{\bar{T}}_{c}})=</math>

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const., i.e.,

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Therefore, eq. (5.297) becomes:

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Therefore, eq. (4) becomes:

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|{{EquationRef|(10)}}

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Since <math>{{{q}''}_{w}}=</math> const, it follows from eq. (5.302) that

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Since <math>{{{q}''}_{w}}=</math> const, it follows from eq. (9) that <math>({{T}_{w}}-{{\bar{T}}_{m}})= Const</math>, i.e.,

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<math>({{T}_{w}}-{{\bar{T}}_{m}})=</math>

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const., i.e.,

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|{{EquationRef|(11)}}

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Combining eqs. (5.300), (5.301) and (5.304), the following relationships are obtained:

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Combining eqs. (7), (8) and (11), the following relationships are obtained:

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The time-averaged mean temperature,

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The time-averaged mean temperature, <math>{{\bar{T}}_{m}}</math>, changes with ''x'' as the result of heat transfer from the tube wall. The rate of mean temperature change can be obtained as follows:

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<math>{{\bar{T}}_{m}}</math>

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, changes with x as the result of heat transfer from the tube wall. By following the same procedure as that in Example 5.2, the rate of mean temperature change can be obtained as follows:

where <math>y={{r}_{0}}-r</math> is the distance measured from the tube wall. Equation (5.307) is subject to the following two boundary conditions:

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where <math>y={{r}_{0}}-r</math> is the distance measured from the tube wall. Equation (14) is subject to the following two boundary conditions:

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Integrating eq. (5.307) in the interval of (r0, r) and considering eq. (5.308), we have:

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Integrating eq. (14) in the interval of (''r<sub>0</sub>'', ''r'') and considering eq. (15), we have:

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Integrating eq. (5.311) in the interval of (0, y) and considering eq. (5.309), one obtains:

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Integrating eq. (18) in the interval of (0, ''y'') and considering eq. (16), one obtains:

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|{{EquationRef|(20)}}

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If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (5.313) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, <math>\bar{u}</math>, in eq. (5.312) can be replaced by <math>{{\bar{u}}_{m}}</math>, and I(y) becomes:

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If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (20) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, <math>\bar{u}</math>, in eq. (19) can be replaced by <math>{{\bar{u}}_{m}}</math>, and ''I''(''y'') becomes:

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|{{EquationRef|(21)}}

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Substituting eqs. (5.314) and (5.306) into eq. (5.313) yields:

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Substituting eqs. (21) and (13) into eq. (20) yields:

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|{{EquationRef|(23)}}

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where y+ is defined in eq. (5.277).

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where ''y''+ is defined as <math>{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }</math>.

In order to obtain the heat transfer coefficient, <math>h={{{q}''}_{w}}/({{T}_{w}}-{{\bar{T}}_{m}})</math>, the temperature difference <math>{{T}_{w}}-{{\bar{T}}_{m}}</math> must be obtained. If the velocity profile can be approximated by eq. (5.287), and the temperature and velocity can also be approximated by the one-seventh law, i.e.,

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|{{EquationRef|(41)}}

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In order to obtain the heat transfer coefficient, <math>h={{{q}''}_{w}}/({{T}_{w}}-{{\bar{T}}_{m}})</math>, the temperature difference <math>{{T}_{w}}-{{\bar{T}}_{m}}</math> must be obtained. If the velocity profile can be approximated by <math>\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}</math>, and the temperature and velocity can also be approximated by the one-seventh law, i.e.,

which can be used together with appropriate friction coefficient discussed in the previous subsection to obtain the Nusselt number.

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which can be used together with appropriate friction coefficient to obtain the Nusselt number.

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==References==

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{{Reflist}}

Current revision as of 20:21, 23 July 2010

Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (q''w = const) will be considered in this article [1][2]. When the turbulent flow in the tube is fully developed, we have and the energy equation becomes

If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (20) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, , in eq. (19) can be replaced by , and I(y) becomes:

In order to obtain the heat transfer coefficient, , the temperature difference must be obtained. If the velocity profile can be approximated by , and the temperature and velocity can also be approximated by the one-seventh law, i.e.,